1914] CALCULATIONS CONCERNING THE MOMENTUM OF PROGRESSIVE WAVES 233
However, a slightly modified form of (3) allows the exponent to be negative. If we take
..... ...................... ...(6)
with /3 positive, we get as above .
f (n \ — '—E^ — rft f"/n\ (p + 1) Cf ,,-s
/ (PO)— - ~a> J (PO) = — -------- :—-.,• ....... ..... (I)
Po PO '
and accordingly P^Po) + _L = izJ? ................ . ........... (8)
& J 4ftJ 2a 4a
If ft = 1, the law of pressure is that under which waves can be propagated without a change . of type, and we see that the momentum is zero. In general, the momentum is positive or negative according as 0 is less or greater than 1.
In the above formula (2) the calculation is approximate only, powers of the disturbance above the second being:.neglected. In the present note it is proposed to determine the sign of the momentum under the laws (3) and (6) more generally and further to extend the calculations to waves in a liquid moving in two dimensions under gravity.
It should be clearly understood that the discussion relates to progressive waves. If this restriction be dispensed with, it would always be, possible to have a disturbance (limited if we please to a finite length) without momentum,, as could be effected very simply by beginning with displacements unaccompanied by velocities. And the disturbance, considered as a whole, can never acquire (or lose) momentum. In order that a wave may be progressive in one direction only, a relation must subsist between the velocity and density at every point. In the case of Boyle's law this relation, first given by De Morgan*,, is
.............................. (9)
and more generally f
„ -
Wherever this relation is violated, a wave emerges travelling in the negative .direction.
For the adiabatic law (3), (10) gives
J ' ...................... '' }
p\ V
.* Ahy, PhiL.Mag, Vol. xxxiv. p, 402 (1849). + Earnshaw, '-P]\il. 'Trans. 1859, p. 146. that (K— 1) R/\ be small, X (equal to 2?r/&) being the wave-length.